Detect Cycle in Linked List Floyd's Algorithm in DSA C - Time & Space Complexity

Time Complexity: Detect Cycle in Linked List Floyd's Algorithm

O(n)

Understanding Time Complexity

We want to understand how long it takes to check if a linked list has a cycle using Floyd's Algorithm.

The question is: how does the time needed grow as the list gets longer?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Detect cycle using Floyd's Algorithm
int hasCycle(Node* head) {
    Node *slow = head, *fast = head;
    while (fast != NULL && fast->next != NULL) {
        slow = slow->next;
        fast = fast->next->next;
        if (slow == fast) {
            return 1; // cycle found
        }
    }
    return 0; // no cycle
}
    

This code moves two pointers through the list at different speeds to find a cycle.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that moves the slow pointer one step and the fast pointer two steps each time.
• How many times: At most, the loop runs proportional to the number of nodes in the list before detecting a cycle or reaching the end.

How Execution Grows With Input

As the list size grows, the number of steps to find a cycle or confirm none grows roughly in a straight line.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: The steps increase directly with the number of nodes, not faster or slower.

Final Time Complexity

Time Complexity: O(n)

This means the time to detect a cycle grows linearly with the list size.

Common Mistake

[X] Wrong: "Because the fast pointer moves twice as fast, the algorithm runs in half the time, so it is O(1)."

[OK] Correct: Even though the fast pointer moves faster, the number of steps still depends on the list length, so the time grows linearly, not constant.

Interview Connect

Understanding this algorithm's time complexity shows you can analyze pointer-based algorithms and reason about loops, a key skill in coding interviews.

Self-Check

"What if we changed the fast pointer to move three steps at a time? How would the time complexity change?"

Practice

1. What is the main purpose of Floyd's Cycle Detection Algorithm in a linked list?

easy

A. To find the middle element of the linked list

B. To detect if there is a cycle (loop) in the linked list

C. To reverse the linked list

D. To sort the linked list in ascending order

Solution

1. Step 1: Understand Floyd's Algorithm goal

   Floyd's Cycle Detection Algorithm is designed to find if a linked list has a cycle or loop.

2. Step 2: Compare with other linked list operations

   Sorting, reversing, or finding the middle element are different operations and not related to cycle detection.

3. Final Answer:

   To detect if there is a cycle (loop) in the linked list -> Option B

4. Quick Check:

   Cycle detection = Detect cycle [OK]

Hint: Floyd's algorithm is famous for cycle detection [OK]

Common Mistakes:

• Confusing cycle detection with sorting
• Thinking it finds middle element
• Assuming it reverses the list

2. Which of the following is the correct way to initialize the two pointers used in Floyd's Cycle Detection Algorithm in C?

easy

A. Node *slow = head; Node *fast = head->next;

B. Node *slow = NULL; Node *fast = head;

C. Node *slow = head->next; Node *fast = head->next->next;

D. Node *slow = head; Node *fast = head;

Solution

1. Step 1: Understand pointer initialization in Floyd's algorithm

   Both slow and fast pointers start at the head node to begin traversal.

2. Step 2: Check options for correct initialization

   Node *slow = head; Node *fast = head; initializes both pointers at head, which is the standard approach.

3. Final Answer:

   Node *slow = head; Node *fast = head; -> Option D

4. Quick Check:

   Both pointers start at head [OK]

Hint: Both pointers start at head node in Floyd's algorithm [OK]

Common Mistakes:

• Starting fast pointer ahead of slow incorrectly
• Initializing slow as NULL
• Skipping initialization of one pointer

3. What will be the output of the following C code snippet using Floyd's algorithm on a linked list with a cycle?

typedef struct Node {
  int data;
  struct Node *next;
} Node;

int detectCycle(Node *head) {
  Node *slow = head, *fast = head;
  while (fast && fast->next) {
    slow = slow->next;
    fast = fast->next->next;
    if (slow == fast) return 1;
  }
  return 0;
}

// Assume linked list has a cycle
printf("%d", detectCycle(head));

medium

A. 1

B. 0

C. Compilation error

D. Infinite loop

Solution

1. Step 1: Analyze the detectCycle function logic

   The function moves slow pointer by one and fast pointer by two nodes. If they meet, it returns 1 indicating a cycle.

2. Step 2: Given the list has a cycle, pointers will meet

   Since the list has a cycle, slow and fast pointers will eventually meet, so function returns 1.

3. Final Answer:

   1 -> Option A

4. Quick Check:

   Cycle present = return 1 [OK]

Hint: Cycle means slow and fast pointers meet, return 1 [OK]

Common Mistakes:

• Assuming function returns 0 for cycle
• Thinking code causes infinite loop
• Confusing return values

4. Identify the bug in this Floyd's cycle detection code snippet:

int detectCycle(Node *head) {
  Node *slow = head, *fast = head;
  while (fast != NULL && fast->next != NULL) {
    slow = slow->next;
    fast = fast->next;
    if (slow == fast) return 1;
    fast = fast->next;
  }
  return 0;
}

medium

A. fast pointer moves two steps but not checked for NULL before second move

B. slow pointer moves two steps instead of one

C. fast pointer moves only one step instead of two

D. slow pointer is not initialized properly

Solution

1. Step 1: Analyze pointer movements in the loop

   Slow moves one step, fast moves one step, then checks if slow == fast, before the second fast move.

2. Step 2: Problem with check timing

   In Floyd's algorithm, fast should move two steps before the cycle check. Here, the check occurs after fast has moved only one step, and the second fast move is done without checking if fast->next is NULL, which can cause a NULL pointer dereference.

3. Final Answer:

   fast pointer moves two steps but not checked for NULL before second move -> Option A

4. Quick Check:

   Second fast move lacks NULL check [OK]

Hint: Fast pointer must complete two steps with NULL checks before cycle check [OK]

Common Mistakes:

• Checking cycle after fast's first step only
• Confusing slow and fast pointer steps
• Not initializing pointers correctly

5. You have a linked list where a cycle exists. After detecting the cycle using Floyd's algorithm, how can you find the starting node of the cycle?

hard

A. Move fast pointer two steps and slow pointer one step until they meet again

B. Count the number of nodes in the list and divide by two

C. Reset slow pointer to head, then move slow and fast one step at a time until they meet

D. Reverse the linked list and check for cycle again

Solution

1. Step 1: Detect cycle and pointers meet inside the loop

   Floyd's algorithm detects cycle when slow and fast pointers meet inside the loop.

2. Step 2: Reset slow to head and move both pointers one step at a time

   Reset slow to head, then move slow and fast one step each until they meet again; this meeting point is the cycle start.

3. Final Answer:

   Reset slow pointer to head, then move slow and fast one step at a time until they meet -> Option C

4. Quick Check:

   Reset slow + move both one step = cycle start [OK]

Hint: Reset slow to head, move both pointers stepwise to find cycle start [OK]

Common Mistakes:

• Moving fast pointer two steps again
• Trying to count nodes to find cycle start
• Reversing list to detect cycle start